Lithium and Beryllium Isotopes with the PAMELA Experiment

The selection criteria applied to each triggered event selected positively charged particles with a precise measurement of the absolute value of the particle rigidity and velocity. The analysis procedure was quite similar to the previous work on the hydrogen and helium isotopes (Adriani et al. 2016), but a few selections had to be modified due to the higher energy loss of lithium and beryllium. Events that were selected required the following:

• 1.  
  A single track fitted within the spectrometer fiducial volume, where the reconstructed track is at least 1.5 mm away from the magnet walls. This analysis employs the position finding algorithm used for the analysis of the boron and carbon spectra (Adriani et al. 2014), which was developed to account for the ionization energy losses that might saturate the read-out electronics for the silicon layers of the tracking system.
• 2.  
  A positive value for the reconstructed track curvature and a positive value for the measured ToF. This selection ensures that the particle enters PAMELA from above.
• 3.  
  Selected tracks must have at least three hits on the x-view and at least two hits on the y-view, for the x-view a "lever arm" (distance between the lowermost and uppermost layer) ≥4 was required. To increase the statistics, these selection criteria are less strict then the ones used in the work on the hydrogen and helium isotopes, where at least four hits on the x-view (bending view) and at least three hits on the y-view were required (Adriani et al. 2016). The full Monte Carlo simulation of the PAMELA apparatus, based on the GEANT4 code (Agostinelli et al. 2003) and described in Adriani et al. (2013b), was used to study the performance of these relaxed tracking criteria. In the rigidity range of this analysis (1–5 GV) no measurable effect was found.
• 4.  
  Charge selection: hydrogen and helium events were identified using the ionization measurements provided by the silicon sensors of the magnetic spectrometer (Adriani et al. 2016) and rejected from the data sample. In the following, lithium and beryllium events were selected by means of ionization energy losses in the ToF system. Each of the 48 channels has been calibrated for a conversion of the dE/dx to a charge value by using the velocity information β from the ToF. The charge for one paddle is then calculated by taking the arithmetic mean of the two PMTs. Finally, charge consistency has been required between S12 (the lower layer of the two layers constituting S1) and $\langle {\rm{S}}2\rangle$ and $\langle {\rm{S}}3\rangle$ (the arithmetic mean of the ionization for the two layers constituting S2 and S3, respectively). In Figure 2 the charge for S12, $\langle {\rm{S}}2\rangle$, and $\langle {\rm{S}}3\rangle$ as a function of the rigidity is shown. The actual selection of Z = 3 or Z = 4 particles is depicted by the solid lines. S11, the upper layer of the two layers constituting S1, was used for efficiency measurements section, see Section 3.4.Note that as β is used to derive the charge, doing the selection as a function of rigidity as shown in Figure 2 has no effect on the isotopic fluxes. Requiring charge consistency above and below the tracking system rejected events interacting in the silicon layers (Adriani et al. 2014).
• 5.  
  Galactic events were selected by imposing that the lower edge of the rigidity bin to which the event belongs exceeds the critical rigidity, ρ_c, defined as 1.3 times the cutoff rigidity ρ_SVC computed in the Störmer vertical approximation.

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Analogous to the analysis described in Adriani et al. (2016), an isotopic separation at fixed rigidity for each sample of Z = 3 and Z = 4 particles is performed. This is possible because the mass of each particle follows the relation m = RZe/γβc (R is the magnetic rigidity, Z × e is the particle charge, and γ is the Lorentz factor). The particle velocity β is provided either directly from the timing measurement of the ToF system, or indirectly from the energy loss in the calorimeter, which follows β via the Bethe–Bloch formula ${dE}/{dx}\propto {Z}^{2}/{\beta }^{2}$ (neglecting logarithmic terms).

For the ToF analysis, the β provided by the timing measurement was used. In Figure 3, β versus the particle rigidity for Z = 3 and Z = 4 data is shown; the black lines in the figure represent the expectations for each isotope.

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The isotopic analysis of nuclei with the calorimeter is restricted to non-interacting events inside the calorimeter. To check if an interaction occurs, events were selected by applying cuts on the ratio Σq_track/Σq_tot in each layer, where q_track is the energy deposited in the strip closest to the track and the neighboring strip on each side, and q_tot the total energy detected in that layer. In case of no interaction, the fraction of q_track/q_tot will be equal to one; if strips outside the track are hit, the value is less then one. The selection used in this work is the same as described in more detail in Adriani et al. (2016): starting from the top of the calorimeter, ${q}_{\mathrm{track}}/{q}_{\mathrm{tot}}$ is calculated at each layer; as long as this value is greater than 0.9, these layers are used for further analysis. A value of 0.9 was chosen because it was found to give a good compromise between high efficiency and rejection of interactions. The dE/dx measurements passing this selection are then sorted and the 50% of samples with larger pulse amplitudes are excluded before taking the mean of the remaining measurements. This truncation method improves the quality of the mean dE/dx value, as in each silicon layer the energy-loss distributions shows a Landau tail, which degrades the resolution of the dE/dx measurement. Finally, an energy-dependent lower limit on the remaining number of layers was used; the selection criteria are the same as in Adriani et al. (2016), requiring at least five measurements at 1 GV, going up to 10 layers at 3 GV. The requirements result in a high selection efficiency with the low energy limit of the analysis of around 200–300 MeV n⁻¹. For more details about different selection criteria and their resulting efficiencies, see Adriani et al. (2016). The selection efficiency of this work will be discussed in Section 3.4

In Figure 4, the mean dE/dx for each event versus the rigidity measured with the magnetic spectrometer for Z = 3 and Z = 4 particles is shown. The energy loss in MeV was derived from the measurement in MIP using a conversion factor (Adriani et al. 2016). In Figures 3 and 4, the isotopic separation between ⁷Be and (⁹Be + ¹⁰Be) is clearly visible, while the separation between ⁶Li and ⁷Li or ⁹Be and ¹⁰Be is difficult to see.

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3.2.1. PAMELA Mass Resolution

To separate isotopes, a good mass resolution is important: for two isotopes with a mass difference of one atomic mass unit (amu) a mass resolution of 0.15 amu (1-sigma) would allow a clear separation of two Gaussian-like distributions without any overlap; a worse mass resolution results in an overlap of the two distributions, but two peaks would still be visible. When the resolution is worse than ≈0.4 amu, it is not possible to resolve the two peaks anymore.

In a magnet spectrometer similar to PAMELA, there are three independent contributions to the mass resolution: the bending power of the magnetic spectrometer coupled with the intrinsic spatial resolution limits of the tracking detectors (MDR_spec); the multiple scattering of the particle along its path in the bending area of the magnet (MDR_coul); and the precision of the velocity measurement (given either by timing or by measuring the energy loss). For PAMELA, the overall momentum resolution of the magnetic spectrometer has been measured in beam tests at CERN for Z = 1 particles (Picozza et al. 2007). From the measurements at high energies, where the contributions from multiple scattering is negligible, one can derive that the MDR_spec of the PAMELA spectrometer has a value of about 1 TV. For Z > 2 particles the tracking algorithm was modified, which resulted in a more precise fitting and a higher track-finding efficiency, but resulting in a worse spatial resolution (Adriani et al. 2014), which reduced the MDR_spec. An analysis of the residuals (the difference between the fitted track and the measured positions) showed that the spatial resolution for lithium particles is comparable to the resolution for protons, while for beryllium particles, the spatial resolution is about twice as large, resulting in an MDR_spec of about 500 GV.

In both cases, the contribution of MDR_spec to the overall mass resolution is negligible for small rigidities (up to a few GV); here, the contribution from multiple scattering is the dominant effect. Its value is inversely proportional to the bending power of the magnet ($\int B\cdot {dl}$) and direct proportional to the amount of matter traversed along the bending part of the track. PAMELA combines a strong magnetic field of 0.43 T with a low amount of material (only the six silicon detectors, each 300 μm, total grammage 0.42 g cm⁻²) in the magnetic cavity. This leads to a respective value for MDR_cou around 3.5% at 8 GV, increasing to about 5% at 1 GV for Z = 1 particles (Picozza et al. 2007), which is in agreement with the GEANT4 PAMELA simulation. For Z = 3 and Z = 4 particles, for which no test beam data were available, the simulation provides values for MDR_cou of ≈3.8% at 5 GV, increasing to about 6%–7% at 1 GV.

In Figure 5, the three independent contributions to the mass resolution for ⁶Li and ⁷Be particle from rigidity (MDR_spec), multiple scattering (MDR_cou), and velocity via ToF are shown as dotted lines, and the red curve shows the overall expected mass resolution of the β_ToF versus rigidity technique. The blue curve displays the overall resolution for the multiple ${dE}/{{dx}}_{\mathrm{Calorimeter}}$ versus rigidity technique simulation (the independent contribution from the calorimeter alone is not shown).

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To derive the mass resolution for flight data, a sample of ⁶Li or ⁷Be particles was selected using strict cuts on the mass coming from the ToF or on the multiple dE/dx from the calorimeter. Thus, the mass resolution of each detector was derived using the other one, assuming that the contamination from other isotopes is negligible. As can be seen, the experimental results on the mass resolution follow the prediction from the GEANT4 PAMELA simulation. Furthermore, the approach with multiple dE/dx from the calorimeter extends the PAMELA measurements on isotopes to higher energies. It can also be noticed that the lower limit of the mass resolution in this analysis is determined by the multiple scattering.

The isotope separation that was performed was identical to that of Adriani et al. (2016) in intervals of kinetic energy per nucleon. As the magnetic spectrometer measures the rigidity of particles, this implies different rigidity intervals according to mass of the isotope under study.

The expected distributions of the measured quantities (dE/dx for the calorimeter and 1/β for the ToF) in each rigidity interval were modeled and then likelihood fits were performed using the "TFractionFitter" method in ROOT (Brun & Rademakers 1997). The GEANT4 PAMELA simulation was used to derive the model distributions for the observables.

As an example, in Figure 6 the distributions of ToF and calorimeter for lithium in the 1.9–2.1 GV rigidity range are shown, and in Figure 7 similar distributions for beryllium. The gray area shows how the combined fit using the two (or three) model distributions derived with the modified GEANT4 PAMELA simulation matches the data points (black points), while the colored areas shows the estimated individual isotope signals.

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3.3.1. Tuning of the Simulation

When analyzing hydrogen and helium isotopes with the calorimeter, taking the simulated energy loss in each layer as coming from the GEANT4 PAMELA simulation, it was noticed that the resulting distributions showed a slight mismatch with the flight data. It was found that the widths of the histograms were smaller than in the real data and there was also a small offset. A multiplicative factor to the simulated energy loss in a layer was applied, plus adding a Gaussian spread of the signal of a few percent (Adriani et al. 2016). These corrections were used also for the analysis of lithium and beryllium. Small deviations were also found for the 1/β distributions of the ToF, which is understandable, as the creation of the timing signals in our simulation is somewhat simplified (for example, no light tracing in the scintillators) compared with the real physical processes.

As one can see from Figure 5, the mass resolution of the simulated data is in good agreement with the flight data, thus the main task was to find the offset between simulated and flight data distributions.

Starting with the beryllium analysis, redundant detectors were used to separate ⁷Be samples from the flight data. As an example, the selection with the calorimeter is shown in Figure 8 on the left, with the actual selection of ⁷Be particles depicted by the red solid lines. To make the selection of low-rigidity events possible, a special truncated mean value of dE/dx was used, taking only the first six layers of the calorimeter into account. It is obvious that the ⁷Be sample will be affected by some contamination from ⁹Be and ¹⁰Be, especially at higher rigidities. Thus, the full simulated data set was created by summing up events from the single ⁷Be, ⁹Be, and ¹⁰Be data sets, with the number of events weighted according to their expected abundance. After applying the same ⁷Be calorimeter selection cuts to the flight data and the simulated data, the 1/β distributions of the selected ⁷Be samples were compared in rigidity intervals. The difference of the two mean 1/β values ("shift") in each rigidity interval is shown in Figure 8 on the right (black dots) together with a best-fit function. The deviation of the fit points from the best-fit curve in Figure 8 is used to estimate the systematic error of the shift function (rms of the spread distribution of the points from the curve).

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The shift curve derived for ⁷Be was then finally used to correct the 1/β values for all the simulated beryllium events, as it is not possible to get the functions for ⁹Be and ¹⁰Be out of the data. This might introduce some systematic error to the calculation of the individual ⁹Be and ¹⁰Be numbers. In a similar manner, the shift function for the truncated mean of the calorimeter was derived, using the ToF to select the ⁷Be sample.

The shift functions for the lithium analysis were derived in the same way as for beryllium, this time using the redundant detectors to select a ⁶Li and ⁷Li sample. ^{As 6}Li and ⁷Li are neighbor isotopes, great care was taken to minimize the effects of the selection and the contamination of one isotope into the other. Finally, the shifts for ⁶Li and ⁷Li were not used individually, but only one best-fit curve was derived for the further analysis, the differences were treated as systematic errors.

As already written, the shift functions were finally used to correct the simulated data event by event, resulting in the distributions shown in Figures 6 and 7.

To derive the isotopic ratios, the numbers of selected events derived with the "TFractionFitter" likelihood procedure had to be corrected for the selections efficiencies and particle losses.

The efficiencies were mostly derived using simulations. These results were then checked with flight data by using redundant detectors to create test samples. For example, the efficiency of the charge selections using S12, $\langle {\rm{S}}2\rangle$, and $\langle {\rm{S}}3\rangle$ was evaluated on a sample of events selected using the charge information of S11 and the calorimeter, similar to the method used for the analysis of boron and carbon fluxes (Adriani et al. 2014). The efficiency of the tracking system was however obtained from the Monte Carlo simulation. The comparison between calorimeter efficiencies derived with simulated data and the ones derived with flight data (using a selection combining the β and dE/dx information from the ToF with the rigidity from the magnetic spectrometer) is shown in Figure 9.

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The calorimeter selection used for this analysis results in a quite flat efficiency of about 50%–60% above 200–300 MeV n⁻¹; for lower energies, the efficiency shows a steep drop, as the particles stop already in the top layers of the calorimeter. As one can see in Figure 9, there is a good agreement between the simulation and flight data for the higher energies. At low energies, the agreement is very good for ⁷Be, while for ⁶Li and ⁷Li there are some differences to see. A similar behavior was found in Adriani et al. (2016) for hydrogen and helium isotopes. It seems that at lower energies, where our selection cuts for the truncated mean are quite soft and only a small number of layers are used to derive the truncated mean, fluctuations are probably still significant, and the GEANT4 PAMELA simulation cannot fully reproduce the actual energy loss under these circumstances. However, the calorimeter selection was used only above 300 MeV n⁻¹. For the calculation of the isotopic ratios the simulated efficiency was used.

Furthermore, one needs the geometrical factor and the live time of the instrument as evaluated by the trigger system (Adriani et al. 2013b), as these values depend on the rigidity. The nominal geometrical factor G_F of PAMELA is almost constant above 1 GV, with the requirements on the fiducial volume corresponding to a value of G_F = 19.9 cm² sr. The live time depends on the orbital selection as described in Section 3.1 and is evaluated by the trigger system (Bruno 2008).

The finite resolution of the magnetic spectrometer and particle slowdown due to ionization energy losses results in a distortion of the particle spectra, which affects the isotopic ratio. We used a Bayesian unfolding procedure (D'Agostini 1995) to correct for this effect (see Adriani et al. 2011).

The particle slowdown in the instrument leads also to a loss of galactic particles, because they do not pass the selection cut in rigidity (ρ > 1.3 ρ_SVC). At 150 MeV n⁻¹, the loss is about 10%, decreasing to just 1% already at 450 MeV n⁻¹. This effect is accounted for using the Monte Carlo simulations.

Because of hadronic interactions in the instrument, lithium and beryllium nuclei might be lost. The energy-dependent correction factor has been derived using the GEANT4 PAMELA simulation. The correction factors at 1 GeV n⁻¹ are ≃17% for lithium and ≃19% for beryllium, with small differences (roughly 1%–3%) between the isotopes.

The selected lithium and beryllium samples are contaminated by secondaries, which are produced in fragmentation processes occurring in the aluminum dome on top of the pressurized vessel. As done in the analysis of the boron and carbon spectra (Adriani et al. 2014), the contamination was studied with a Monte Carlo calculation based on the FLUKA code (Battistoni et al. 2007). Cosmic spectra for oxygen, carbon, boron, and beryllium were simulated and then scaled to the expected numbers at the top of the instrument. We found that roughly 1.5 times more ⁷Li than ⁶Li is produced, but because the number of produced ⁶Li or ⁷Li is less than 1% of the selected flight data sample, the effect on the ⁷Li/⁶Li ratio is negligible. The fragmentation of oxygen and carbon into beryllium creates roughly the same numbers of the three beryllium isotopes; however, the fragmentation of ¹⁰B and ¹¹B creates significantly more ⁹Be and ¹⁰Be than ⁷Be. After subtracting the contamination, the ⁷Be/(⁹Be + ¹⁰Be) ratio increases by ≃7% at 200 MeV n⁻¹, while the difference is ≃3% at 1 GeV n⁻¹.

Systematic uncertainties were estimated as discussed in Adriani et al. (2016). The event selection criteria described in Section 3.1 are less strict compared with our previous works (Adriani et al. 2011, 2016), where we quoted a systematic uncertainty of 3.6%. However, it was found in Adriani et al. (2013a) that less-stringent cuts implied a larger systematic error. For this work, a systematic error of 5% was derived.

The correction for particles lost due to the selection of galactic particles (see Section 3.1) has an uncertainty due to the size of the Monte Carlo sample. The systematic error decreases from 3% at 100 MeV n⁻¹ to 0.05% at 1000 MeV n⁻¹.

Using the GEANT4 PAMELA simulation, it was studied how the uncertainties on the modeled distribution affects the reconstructed number of events. It was found that the dominant effect was the shifts of the modeled distributions. For example, the systematic uncertainty of the shift for the truncated mean of the calorimeter transforms in a systematic error of the measured ⁶Li or ⁷Li numbers of roughly 20%.

The efficiency of the calorimeter selection was derived using simulated and flight data, as shown in Figure 9. The agreement between the two methods is quite good, and we assigned a conservative systematic error of 3% independent from the energy.

The systematic uncertainties of the unfolding procedure (2%, independent of energy) and the effective geometrical factor (0.18%, practically independent of energy) are taken from Adriani et al. (2016) without changes, and we refer the reader to this paper for more details.

The uncertainty of the correction for particles lost due inelastic interactions is ≃1% due to the size of the Monte Carlo sample.